Mess [1]} proved that when \(M\)  is a quasifuchsian AdS spacetime with Cauchy surface a closed surface \(S\) , its holonomy representation \(\rho :\pi _1(S)\rightarrow SO_0(2,2)\)  can be written, in the decomposition of \(SO_0(2,2)\)  as \(PSL(2,{\mathbb {R}})\times PSL(2,{\mathbb {R}})\) , as \(\rho =(\rho _L, \rho _R)\) , where \(\rho _L,\rho _R:\pi _1(S)\rightarrow PSL(2,{\mathbb {R}})\)  have maximal Euler number, and are therefore (by a result of Goldman [2]}) holonomy representations of hyperbolic metrics on \(S\) , which can be called the left and right hyperbolic metrics of \(M\) .
